Rough Energy Landscape Research Articles

Order-parameter-aided temperature-accelerated sampling for the exploration of crystal polymorphism and solid-liquid phase transitions.

The problem of predicting polymorphism in atomic and molecular crystals constitutes a significant challenge both experimentally and theoretically. From the theoretical viewpoint, polymorphism prediction falls into the general class of problems characterized by an underlying rough energy landscape, and consequently, free energy based enhanced sampling approaches can be brought to bear on the problem. In this paper, we build on a scheme previously introduced by two of the authors in which the lengths and angles of the supercell are targeted for enhanced sampling via temperature accelerated adiabatic free energy dynamics [T. Q. Yu and M. E. Tuckerman, Phys. Rev. Lett. 107, 015701 (2011)]. Here, that framework is expanded to include general order parameters that distinguish different crystalline arrangements as target collective variables for enhanced sampling. The resulting free energy surface, being of quite high dimension, is nontrivial to reconstruct, and we discuss one particular strategy for performing the free energy analysis. The method is applied to the study of polymorphism in xenon crystals at high pressure and temperature using the Steinhardt order parameters without and with the supercell included in the set of collective variables. The expected fcc and bcc structures are obtained, and when the supercell parameters are included as collective variables, we also find several new structures, including fcc states with hcp stacking faults. We also apply the new method to the solid-liquid phase transition in copper at 1300 K using the same Steinhardt order parameters. Our method is able to melt and refreeze the system repeatedly, and the free energy profile can be obtained with high efficiency.

Disorder, exchange and magnetic anisotropy in the room-temperature molecular magnet V[TCNE]x – A theoretical study

We report quantum chemical calculations to address yet unresolved and puzzling questions regarding the structural and magnetic disorder of V[TCNE]x (TCNE=tetracyanoethylene, x∼2), the first room-temperature molecule-based magnet. Starting from an ideal lattice model, containing TCNE ligands either tetra- or bi-connected to vanadium(II) ions, we identify the key sources of structural disorder, explaining the amorphousness and non-stoichiometric nature of V[TCNE]x. The proposed model is prone to static disorder in terms of the bulk distribution of the tetra-connected TCNE species and to dynamic effects due to the relative rotational freedom of the bi-connected TCNE moieties. Density functional theory (DFT) calculations of the model system with rotated TCNE molecules show a rough energy landscape, consistent with the presence of magnetic irreversibilities in the system. The broken symmetry DFT approach evidences ferrimagnetic spin orientation for all TCNE configurations, ruling out the spin glass model. Multiconfigurational calculations with additional spin–orbit interaction allow for the account of the single-ion-anisotropy of the V(II) ions in different environments. We determine a small uniform zero-field-splitting (Dc=−0.03K) of the bulk as well as a sizeable random anisotropy (Dr=0.56K) due to TCNE vacancies. We clarify the interplay of ferrimagnetism and random magnetic anisotropy in this system, which favours correlated sperimagnetic and not spin glass behaviour, in agreement with puzzling experimental data. Our approach goes beyond the material of interest here, as it can be applied to other disordered molecular magnets by correlating the sources of disorder with their effects on the magnetic properties.

Robust Estimation of Diffusion-Optimized Ensembles for Enhanced Sampling

Molecular dynamics and Monte Carlo simulation can provide detailed structural insights into biological systems by exploring the distribution of accessible conformational states. However, due to the rough free energy landscapes characterizing many systems, obtaining converged statistics in such simulation remains an important challenge. In order to address this issue, over the past decades, a number of enhanced sampling strategies have been proposed, including methods like simulated annealing, replica exchange, metadynamics and generalized ensemble approaches such as multicanonical (flat histogram) sampling. Most of these methods were designed to soften the landscape, making it easier to overcome the free energy barriers. Recently, it was proposed that additional performance gains could be obtained by taking the position-dependent diffusion coefficient along the reaction coordinate into account, thus placing greater emphasis on regions diffusing slowly so that the round trip rate is maximized (Trebst et al. 2004. 10.1103/PhysRevE.70.046701). Although the diffusion-optimized ensemble approach has shown great promise, the practical application of it has been hindered by the serious challenge of estimating the diffusion coefficient accurately, especially at the preliminary stage of a simulation when only a small part of the phase space has been explored. In this study, a simple yet robust solution to this problem is presented. We propose replacing the standard global estimation procedure with a local estimation technique that was previously introduced in the context of trajectory analysis and reaction coordinate optimization (Krivov et al. 2008. 10.1073/pnas.0800228105). We demonstrate that the precision of the proposed estimation method is dramatically increased, requiring considerable fewer data compared to current state-of-the-art procedures. Finally, we show that with our method, diffusion-optimized sampling arises as a natural generalization of the standard multicanonical ensemble, estimating a simple histogram of transitions rather than the usual histogram of states.

Accelerate Sampling in Atomistic Energy Landscapes using Topology-Based Coarse-Grained Models

Successful simulations of biomolecules require sufficient sampling of the relevant conformational space. This has remained a major challenge due to large conformational space and significant energy barriers, especially with atomistic force fields. Temperature replica exchange (REX) has emerged as a powerful and popular technique for enhanced sampling. Yet, efficiency of temperature REX can be severely limited by the presence of sharp cooperative conformational transitions as well as due to large entropic barriers frequently associated with folding transitions. A coarse-grained representation is often necessary to significantly reduce the conformational space and allows faster sampling of reversible conformational transitions, albeit at the expense of reduced detail and accuracy. Here, we describe a multi-scale enhanced sampling (MSES) method that directly couples topology-based coarse-grained protein models with atomistic ones to accelerate the sampling of complex and rough atomistic energy landscapes. The bias from the coupling potential is completely removed by performing Hamiltonian/temperature REX, allowing one to benefit simultaneously from faster transitions of the coarse-grained model and the accuracy of the atomistic force field. The method has been applied to implicit solvent simulations of several peptides and small proteins including protein GB1, protein A and villin headpiece. The results demonstrate that MSES dramatically increases the number of folding/unfolding transitions sampled and improve the convergence of various thermodynamic properties of interest in all cases. Importantly, this method is simple and fully scalable to larger and more complex systems.

Temperature Integration: An efficient procedure for calculation of free energy differences

We propose a method, Temperature Integration, which allows an efficient calculation of free energy differences between two systems of interest, with the same degrees of freedom, which may have rough energy landscapes. The method is based on calculating, for each single system, the difference between the values of lnZ at two temperatures, using a Parallel Tempering procedure. If our two systems of interest have the same phase space volume, they have the same values of lnZ at high-T, and we can obtain the free energy difference between them, using the two single-system calculations described above. If the phase space volume of a system is known, our method can be used to calculate its absolute (versus relative) free energy as well. We apply our method and demonstrate its efficiency on a “toy model” of hard rods on a 1-dimensional ring.

Weak adhesion at the mesoscale: particles at an interface

Adhesion of colloidal particles (synthetic or biological) to one another or to other surfaces, typically immersed in a fluid medium, is frequently a mesoscale phenomenon, intermediate between microscopic and macroscopic scales. Similar to other mesoscale phenomena, gaining a useful understanding based on ideas from either continuum or atomistic concepts alone is limited. Combination of experimental techniques like optical trapping and scanning probe techniques with simultaneous imaging provides direct insight into the space- and time-dependent dynamics of the adhesion process in the relevant scales. The strength of adhesion typically evolves with time; the details of the processes depend on the overall strength of attractive interaction among the adhering objects as well as its spatial fluctuations governed by the randomness of the surfaces involved and their own time-dependent reconstruction due to varying local stresses. This review focuses on situations where the adhesive inter-object interactions are “weak” compared to the cohesive intra-object interactions whereby all processes occur, to a good approximation, in the interfacial region. In these situations, the dynamics of adhesion is affected by both the “quenched” spatial disorder of the surfaces as well as the thermal fluctuations. A local microrheological characterization of the stress-coupling in the interfacial region provides a heuristic description of the adhesion process. The discrete, thermally assisted “punctuated descent” of the effective adhering system down a rough energy landscape through a hierarchy of finite metastable minima reflects the mesoscopic nature of adhesion. We review these experimental observations of particle adhesion in the weak limit and explore possible common mechanisms underlying apparently disparate phenomena and processes.

Characterization of Viscous and Excluded Volume effects on the Folding Kinetics of the Tetraloop-Receptor Motif: Single Molecule Tests of Kramer's Theory

It is widely known that there exists a strong relationship between the solvent, including co-solutes (e.g. cations, proteins, nucleic acids), and RNA that cooperatively defines the kinetics of tertiary structure formation. Viscosity, a solvent property that is coupled to dynamics, is critical in RNA folding due to the propensity for populating kinetically trapped species on a rough free energy landscape. Also of importance is the role of macromolecular co-solutes that limit the volume available for molecules to sample (excluded volume effects). The effects of these properties are highlighted by observed in vivo diffusion coefficients that are 4-200x smaller than in aqueous solvents. In this work, Kramers' rate theory is used to describe the viscosity dependence of tetraloop-receptor docking kinetics at the single molecule level. Both rate constants, kfold and kunfold, decrease with increasing viscosity (increasing glycerol %), trends that are predicted by Kramers' theory in the over-damped limit. However, the same measurements made in high molecular weight PEG solutions showed folding rate constants are accelerated by 1-2 orders of magnitude. Scaled particle theory, describing a hard spheres PEG-RNA interaction, quantitatively predicts the stabilizing effect of excluded volume. Temperature dependent measurements show that the thermodynamics of docking are not perturbed (ΔH°= −23(2) kcal/mol and ΔS°= −76(6) cal/mol∗K) even in up to 50% glycerol (∼6 cP) and that solvent activation accounts for 4-6 kcal/mol of the folding enthalpy. In PEG solutions thermodynamics reveal that the folding is stabilized by a reduction in the entropy of the unfolded RNA ensemble (ΔΔS°>0). Thus, these studies validate physical models describing characteristic kinetic and thermodynamic trends for intramolecular structure formation attributed to viscosity and excluded volume.

Protein Folding: Adding a Nucleus to Guide Helix Docking Reduces Landscape Roughness

The elongated three-helix‐bundle spectrin domains R16 and R17 fold and unfold unusually slowly over a rough energy landscape, in contrast to the homologue R15, which folds fast over a much smoother, more typical landscape. R15 folds via a nucleation–condensation mechanism that guides the docking of the A and C-helices. However, in R16 and R17, the secondary structure forms first and the two helices must then dock in the correct register. Here, we use variants of R16 and R17 to demonstrate that substitution of just five key residues is sufficient to alter the folding mechanism and reduce the landscape roughness. We suggest that, by providing access to an alternative, faster, folding route over their landscape, R16 and R17 can circumvent their slow, frustrated wild-type folding mechanism.

Conformational Transition and Energy Landscape of ErbB4 Activated by Neuregulin1β: One Microsecond Molecular Dynamics Simulations

ErbB4, a receptor tyrosine kinase of the ErbB family, plays crucial roles in cell growth and differentiation, especially in the development of the heart and nervous system. Ligand binding to its extracellular region could modulate the activation process. To understand the mechanism of ErbB4 activation induced by ligand binding, we performed one microsecond molecular dynamics (MD) simulations on the ErbB4 extracellular region (ECR) with and without its endogenous ligand neuregulin1β (NRG1β). The conformational transition of the ECR-ErbB4/NRG1β complex from a tethered inactive conformation to an extended active-like form has been observed, while such large and function-related conformational change has not been seen in the simulation on the ECR-ErbB4, suggesting that ligand binding is indeed the active inducing force for the conformational transition and further dimerization. On the basis of MD simulations and principal component analysis, we constructed a rough energy landscape for the conformational transition of ECR-ErbB4/NRG1β complex, suggesting that the conformational change from the inactive state to active-like state involves a stable conformation. The energy barrier for the tether opening was estimated as ~2.7 kcal/mol, which is very close to the experimental value (1-2 kcal/mol) reported for ErbB1. On the basis of the simulation results, an atomic mechanism for the ligand-induced activation of ErbB4 was postulated. The present MD simulations provide a new insight into the conformational changes underlying the activation of ErbB4.

Mechanical Behavior of Non-bonded Fiber Networks in Compression

The mechanical behavior of an ensemble of fibers which are not bonded to each other subjected to compression is studied using a numerical model. It is shown that the stress-strain curve is a power law and the strain energy is stored predominantly in the bending deformation mode of the fibers. Hysteresis is observed when the system is subjected to cyclic loading-unloading, with a stable hysteresis loop being obtained after few cycles. The deformation is not smooth; rather it is dominated by instabilities which lead to pronounced stress drops. These instabilities are due to relative fiber sliding and rearrangement and a correlation is observed between stress drops and collective sliding events. The resulting jerky flow is observed even in absence of friction between fibers and is a result of the evolution of the system over a rough energy landscape during compression.

A Model with Two Elastically Coupled Reaction Coordinates for the Internal Friction of Proteins

The rates of protein conformational changes are usually not only limited by external but also internal friction, however, the origin and significance of this latter phenomenon is poorly understood. It is often found experimentally that a linear fit to the reciprocal of the reaction rate as a function of the viscosity of the external medium has a non-zero value at zero viscosity, signifying the presence of internal friction. Furthermore, some of the experiments performed at different temperatures indicate that the internal friction follows an Arrhenius-like temperature dependence. To explain these phenomena we suggest a simple model for protein conformational changes in terms of two elastically coupled reaction coordinates, one of which being in contact with the external medium, and the other one experiencing the rough energy landscape of the protein. Our analytical calculations, supplemented with numerical simulations demonstrate that depending on the coupling strength (which is related to the flexibility of the protein) the short-wavelength components of the energy landscape roughness can be observed as an increased apparent internal friction with an apparent activation energy.

Unraveling the Beautiful Complexity of Simple Lattice Model Polymers and Proteins Using Wang-Landau Sampling

We describe a class of “bare bones” models of homopolymers which undergo coil-globule collapse and proteins which fold into their native states in free space or into denatured states when captured by an attractive substrate as the temperature is lowered. We then show how, with the use of a properly chosen trial move set, Wang-Landau Monte Carlo sampling can be used to study the rough free energy landscape and ground (native) states of these intriguingly simple systems and thus elucidate their thermodynamic complexity.

Monte Carlo Methods for Rough Free Energy Landscapes: Population Annealing and Parallel Tempering

Parallel tempering and population annealing are both effective methods for simulating equilibrium systems with rough free energy landscapes. Parallel tempering, also known as replica exchange Monte Carlo, is a Markov chain Monte Carlo method while population annealing is a sequential Monte Carlo method. Both methods overcome the exponential slowing associated with high free energy barriers. The convergence properties and efficiency of the two methods are compared. For large systems, population annealing initially converges to equilibrium more rapidly than parallel tempering for the same amount of computational work. However, parallel tempering converges exponentially and population annealing inversely in the computational work so that ultimately parallel tempering approaches equilibrium more rapidly than population annealing.

Multi-Scale Free Energy Landscape calculation method by combination of coarse-grained and all-atom models

We propose a new multi-scale molecular dynamics simulation method for calculating free-energy landscapes of biomolecular systems, namely the Multi-Scale Free Energy Landscape calculation method (MSFEL). This method is designed to simultaneously achieve high accuracy and sampling efficiency using a combination of coarse-grained and all-atom models. First, the coarse-grained model is used for calculation of a rough energy landscape over a broad conformational space. Subsequently the all-atom model is used to increase the sampling accuracy of local energy landscapes. MSFEL was applied to Met-enkephalin and Trpzip2, and was demonstrated to be more efficient in conformational sampling compared to conventional molecular dynamics.

Improving the Wang–Landau algorithm for polymers and proteins

The 1/t Wang-Landau algorithm is tested on simple models of polymers and proteins. It is found that this method resolves the problem of the saturation of the error present in the original algorithm for lattice polymers. However, for lattice proteins, which have a rough energy landscape with an unknown energy minimum, it is found that the density of states does not converge in all runs. A new variant of the Wang-Landau algorithm that appears to solve this problem is described and tested. In the new variant, the optimum modification factor is calculated in the same straightforward way throughout the simulation. There is only one free parameter for which a value of unity appears to give near optimal convergence for all run lengths for lattice homopolymers when pull moves are used. For lattice proteins, a much smaller value of the parameter is needed to ensure rapid convergence of the density of states for energies discovered late in the simulation, which unfortunately results in poor convergence early on in the run.

Population annealing with weighted averages: A Monte Carlo method for rough free-energy landscapes

The population annealing algorithm introduced by Hukushima and Iba is described. Population annealing combines simulated annealing and Boltzmann weighted differential reproduction within a population of replicas to sample equilibrium states. Population annealing gives direct access to the free energy. It is shown that unbiased measurements of observables can be obtained by weighted averages over many runs with weight factors related to the free-energy estimate from the run. Population annealing is well suited to parallelization and may be a useful alternative to parallel tempering for systems with rough free-energy landscapes such as spin glasses. The method is demonstrated for spin glasses.

Heat Exchange between Two Interacting Nanoparticles beyond the Fluctuation-Dissipation Regime

We show that the observed nonmonotonic behavior of the thermal conductance between two nanoparticles when they are brought into contact is originated by an intricate phase space dynamics. Here it is assumed that this dynamics results from the thermally activated jumping through a rough energy landscape. A hierarchy of relaxation times plays the key role in the description of this complex phase space behavior. Our theory enables us to analyze the heat transfer just before and at the moment of contact.

Phospholipid Reorientation at the Lipid/Water Interface Measured by High Resolution 31P Field Cycling NMR Spectroscopy

The magnetic field dependence of the 31P spin-lattice relaxation rate, R1, of phospholipids can be used to differentiate motions for these molecules in a variety of unilamellar vesicles. In particular, internal motion with a 5- to 10-ns correlation time has been attributed to diffusion-in-a-cone of the phosphodiester region, analogous to motion of a cylinder in a liquid hydrocarbon. We use the temperature dependence of 31P R1 at low field (0.03–0.08 T), which reflects this correlation time, to explore the energy barriers associated with this motion. Most phospholipids exhibit a similar energy barrier of 13.2 ± 1.9 kJ/mol at temperatures above that associated with their gel-to-liquid-crystalline transition (Tm); at temperatures below Tm, this barrier increases dramatically to 68.5 ± 7.3 kJ/mol. This temperature dependence is broadly interpreted as arising from diffusive motion of the lipid axis in a spatially rough potential energy landscape. The inclusion of cholesterol in these vesicles has only moderate effects for phospholipids at temperatures above their Tm, but significantly reduces the energy barrier (to 17 ± 4 kJ/mol) at temperatures below the Tm of the pure lipid. Very-low-field R1 data indicate that cholesterol inclusion alters the averaged disposition of the phosphorus-to-glycerol-proton vector (both its average length and its average angle with respect to the membrane normal) that determines the 31P relaxation.

Biomolecule Association Rates Do Not Provide a Complete Description of Bond Formation

The efficiency of many cell-surface receptors is dependent on the rate of binding soluble or surface-attached ligands. Much effort was exerted to measure association rates between soluble molecules (three-dimensional kon) and, more recently, between surface-attached molecules (two-dimensional [2D] kon). According to a generally accepted assumption, the probability of bond formation between receptors and ligands is proportional to the first power of encounter duration. Here we provide new experimental evidence and review published data demonstrating that this simple assumption is not always warranted. Using as a model system the (2D) interaction between ICAM-1-coated surfaces and flowing microspheres coated with specific anti-ICAM-1 antibodies, we show that the probability of bond formation may scale as a power of encounter duration that is significantly higher than 1. Further, we show that experimental data may be accounted for by modeling ligand-receptor interaction as a displacement along a single path of a rough energy landscape. Under a wide range of conditions, the probability that an encounter of duration t resulted in bond formation varied as erfc[(t0/t)1/2], where t0 was on the order of 10 ms. We conclude that the minimum contact time for bond formation may be a useful parameter to describe a ligand-receptor interaction, in addition to conventional association rates.

DNA energy landscapes via calorimetric detection of microstate ensembles of metastable macrostates and triplet repeat diseases

Biopolymers exhibit rough energy landscapes, thereby allowing biological processes to access a broad range of kinetic and thermodynamic states. In contrast to proteins, the energy landscapes of nucleic acids have been the subject of relatively few experimental investigations. In this study, we use calorimetric and spectroscopic observables to detect, resolve, and selectively enrich energetically discrete ensembles of microstates within metastable DNA structures. Our results are consistent with metastable, "native" DNA states being composed of an ensemble of discrete and kinetically stable microstates of differential stabilities, rather than exclusively being a single, discrete thermodynamic species. This conceptual construct is important for understanding the linkage between biopolymer conformational/configurational space and biological function, such as in protein folding, allosteric control of enzyme activity, RNA and DNA folding and function, DNA structure and biological regulation, etc. For the specific DNA sequences and structures studied here, the demonstration of discrete, kinetically stable microstates potentially has biological consequences for understanding the development and onset of DNA expansion and triplet repeat diseases.